Domain walls, near-BPS bubbles, and probabilities in the landscape

We develop a theory of static Bogomol’nyi-Prasad-Sommerfield (BPS) domain walls in stringy landscape and present a large family of BPS walls interpolating between different supersymmetric vacua. Examples include Kachru, Kallosh, Linde, Trivedi models, STU models, type IIB multiple flux vacua, and models with several Minkowski and anti-de Sitter vacua. After the uplifting, some of the vacua become de Sitter (dS), whereas some others remain anti-de Sitter. The near-BPS walls separating these vacua may be seen as bubble walls in the theory of vacuum decay. As an outcome of our investigation of the BPS walls, we found that the decay rate of dS vacua to a collapsing space with a negative vacuum energy can be quite large. The parts of space that experience a decay to a collapsing space, or to a Minkowski vacuum, never return back to dS space. The channels of irreversible vacuum decay serve as sinks for the probability flow. The existence of such sinks is a distinguishing feature of the landscape. We show that it strongly affects the probability distributions in string cosmology.

Figure 1

KKLT potential $V$ (blue upper curve) and $\mathcal{Z}$ (red curve below) as functions of the modulus. One can see that the potential has an AdS minimum at $\varphi ={\varphi }^{*}\simeq 4.09854$ and tends to Minkowski limit at large $\varphi$. $\mathcal{Z}$ is always negative in the interval between AdS and Minkowski vacua. Potential energy density $V$ is shown in units ${10}^{-15}{M}_p^4$, whereas the covariantly holomorphic superpotential $\mathcal{Z}$ is shown in units ${10}^{-8}{M}_p^3$.

Figure 2

The KKLT wall interpolating between the AdS and flat Minkowski space which are at infinite distance from each other in the moduli space. We plot the scalar $\varphi$, the covariantly holomorphic superpotential $\mathcal{Z}$ (in units ${10}^{-9}$), the warp factor $a$ and the curvature $R$ (in units ${10}^{-15}$), all as functions of $r$. Here $r$ is the coordinate of the domain wall configuration given in Eq. (2.6); we show $r$ in units ${10}^7$.

Figure 3

Here is the contour plot of the covariantly holomorphic superpotential $\mathcal{Z}({\varphi }_1,{\varphi }_2)$ and the gradient flow between the two critical points. $\mathcal{Z}({\varphi }_1,{\varphi }_2)$ is everywhere positive between AdS critical points, it vanishes only at the asymptotically Minkowski vacuum.

Figure 4

Flow of the potential $V({\varphi }_1(r),{\varphi }_2(r))$.

Figure 5

Flow on $\psi$ complex plane.

Figure 6

Flow on $\tau$ complex plane.

Figure 7

Flow of the potential $V(\psi (u),\tau (u))$.

Figure 8

The potential energy density $V$, blue upper curve, and the covariantly holomorphic superpotential $\mathcal{Z}$, the red curve below, as the functions of the modulus $\varphi$ for the KL model. The potential is shown in units ${10}^{-8}$, and $\mathcal{Z}$ is in units ${10}^{-4}$. Both $V$ and $\mathcal{Z}$ have extrema at the same point, which is a reflection of the fact that supersymmetry is unbroken in the minima of $V$. This plot is for $A=1$, $B=-2$, $W_0=-0.125$, $a=\pi /100$ and $b=\pi /50$.

Figure 9

KL wall between $M_1$ and AdS. We plot the scalar $\varphi$, $\mathcal{Z}$ (in units ${10}^{-5}$), warp factor $a$ and the curvature scalar $R$ (in units ${10}^{-8}$) as functions of the coordinate $r$. The curvature $R$ is equal to zero in the Minkowski vacuum, has a peak at the domain wall, and approaches a constant negative value in the AdS vacuum.

Figure 10

The potential $V$ (upper curve, in units ${10}^{-8}$) and the covariantly holomorphic superpotential $\mathcal{Z}$ (lower curve, in units ${10}^{-4}$) for the KL model, as function of the modulus. Notice that we took $A=0.96$ instead of $A=1$, all other parameters being the same as for the potential in Fig. 8.

Figure 11

KL wall with a critical tension between ${\mathrm{AdS}}_1$ and ${\mathrm{AdS}}_2$. We plot the scalar $\varphi$ and the covariantly holomorphic superpotential $\mathcal{Z}$ (in units ${10}^{-5}$), which is always negative, warp factor $a$, and the curvature scalar $R$ (in units ${10}^{-8}$), as a function of the coordinate $r$. The curvature $R$ has a peak at the domain wall, and approaches different constant negative values in two different AdS minima.

Figure 12

The potential $V$ (upper curve, in units ${10}^{-8}$) and the covariantly holomorphic superpotential $\mathcal{Z}$ (lower curve, in units ${10}^{-4}$) for the KL model as function of the modulus. $\mathcal{Z}$ is positive at ${\varphi }_0$ and negative at ${\varphi }_1$. Here we took $A=1.05$ instead of $A=1$, all other parameters being the same as for the potential in Fig. 8.

Figure 13

A BPS wall between two AdS domains in the version of the KL model where $\mathcal{Z}$ changes the sign on the domain wall solution. We plot the scalar $\varphi$, the covariantly holomorphic superpotential $\mathcal{Z}$ (in units ${10}^{-5}$), the warp factor $a$, and the curvature $R$ (in units ${10}^{-8}$). Notice the characteristic peak of the scale factor at the point where $\mathcal{Z}$ changes sign.

Figure 14

A slightly uplifted KL potential, shown in units ${10}^{-12}$.